# proof that all cyclic groups are abelian

The following is a proof that all cyclic groups are abelian.

###### Proof.

Let $G$ be a cyclic group^{} and $g$ be a generator^{} of $G$. Let $a,b\in G$. Then there exist $x,y\in \mathbb{Z}$ such that $a={g}^{x}$ and $b={g}^{y}$. Since $ab={g}^{x}{g}^{y}={g}^{x+y}={g}^{y+x}={g}^{y}{g}^{x}=ba$, it follows that $G$ is abelian^{}.
∎

Title | proof that all cyclic groups are abelian |
---|---|

Canonical name | ProofThatAllCyclicGroupsAreAbelian |

Date of creation | 2013-03-22 13:30:44 |

Last modified on | 2013-03-22 13:30:44 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 7 |

Author | Wkbj79 (1863) |

Entry type | Proof |

Classification | msc 20A05 |